If you want to learn about general relativity, there are unfortunately few alternatives other than very complicated textbooks or vague dumbed-down popularization that explains nothing... In this post, we will try to find a way in the middle.
We will embark upon a fascinating voyage into the heart of differential geometry, and discover the inner workings of the theory of general relativity and Einstein's equation.
With visual represtentations of the Riemann tensor, the Ricci tensor and of the notion of connection !...

Updated (12/09/2016). The (foolish) purpose of this post is to tackle the concept of tensor, while trying to keep it accessible to the widest audience possible. For the layman (or specialist from another field of study) who just wants to know what this is about without the rigidity and complexity of reading an indecipherable math handbook, for the student who already had a course on tensor calculus but can't figure out some of the "obvious" stuff the teacher didn't linger on, and finally for any math teacher or expert looking for a fresh eye on the subject (meaning not the handbook type) maybe to help describe it and teach it to his own audience....

If you are here, then it's because you don't like formulas and complicated calculations too much, but you're still curious about what is a tensor.
So we will do a quick tour of the concept, in the easiest way possible....

« Eternity is really long, especially near the end. » Woody Allen (1935 - )
What is infinity ? How to grasp such a concept, by definition so immense that nothing can contain it ?
As surprising as it may seem, the answer lies not in philosophy, but really in mathematics.
We will see that it is possible to define infinity in a rigorous and coherent manner. There even exists numerous different infinities : precisely an infinity of them !...

In 1959 was published in the journal Pride of the American College Public Relations Association an essay entitled Angels on a Pin, by Alexander Calandra, professor of physics at Washington University in St. Louis, Missouri. The story is about a physics student who surprises his professor on a simple question of physics....

« The most desperate are the most beautiful lament »
(Alfred de Musset, La nuit de Mai, 1835)
A mathematician’s lament (2002) is an article from Paul Lockhart, a first-class research mathematician and teacher at Saint Ann’s School in Brooklyn, New York. Not only is it a critique of current K-12 mathematics education in the United States, but it’s also a reminder that mathematics are also and above all an art, like music or beaux-arts. I invite you to read at least the two introductory pages....

A quantity of -2 objects, a marathon runner finishing -3rd, numbers smaller than nothing ?! What do negative numbers mean ? The answer to that question is easy only to those who are used to them. And for most of us, it doesn't matter what they really are or what they mean, because the thing is these numbers are quite useful !...

The Icosien game is a graph theory game. It was invented in 1857 by Sir W.R.Hamilton (1805-1865), a great mathematician to whom we owe - among other things - a reformulation of mechanics' formalism which now bears his name, and quaternions (which we'll tackle in a future article).
In this post, we will focus on a flash version of this game, written by Neamar. More precisely, we will see the mathematical principles behind the game and a method for solving the last two levels.
ATTENTION SPOILER ALERT !
If you don't know this game, don't read this article right now, but try the game beforehand !...

« Put your hand on a stove for a minute and it will feel like an hour. Sit with a pretty girl for an hour and it will fly like a minute. That’s relativity. »
This famous quote, attributed to Einstein 1 is a good description of the psychological side of time. Time is undeniably linked to our senses, we perceive it through duration, order and simultaneity. What will be left of it if we take those away ?...

« God made the integers, all the rest is the work of man » Léopold Kronecker (1823-1891). There is no satisfying definition of the general concept of number. However, lots of particular numbers can be rigorously defined. Natural numbers, integers, imaginary, transcendent, algebraic, computable numbers, etc. In this "story", we will see examples of numbers and will try to understand them intuitively and visually. Some of these numbers will not even look at all like numbers... But this first episode is about natural numbers. That's a good start....

« We are like dwarfs on the shoulders of giants. »
This famous metaphor, attributed to Bernard de Chartres, a XIIth century philosopher, reused by Newton and Pascal among others, is a tribute to savant predecessors and an acknowledgment of the cumulative nature of scientific knowledge.
In this article, we will pay a tribute to Henri Poincaré, a brilliant mathematician, universal thinker and remarkable physicist. However, we will not try to grant him what is not his, but we will acknowledge some of his numerous contributions to the theory of relativity, which main idea is clearly owed to Albert Einstein....

In its modern interpretation, the principle of relativity is profoundly linked to the group structure of Lorentz transformations.
We will describe the equivalence relation between the two, and at the same time give a geometrical description of what is an inertial reference frame in special relativity....

« Philosophy is written in this grand book - I mean the Universe - which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles and other geometrical figures, without which it is humanly impossible to understand a single word of it. » G. Galilei, Il Saggiatore (The Assayer), 1623....

I suggest a little pseudo-scientific experiment which goal is to literally picture how useful the above mentioned study can be.
In this so-called experiment, we will inquire about the daily activities of a human being. Let's consider an ordinary human being, from any ethnic origin or religion, etc. Then any resemblance to one of your relatives or friends would be purely coincidental. Let's name this human being Mr A. Conveniently a very common letter....